Abstract
Sachdev-Ye-Kitaev (SYK) model has emerged as a new paradigm of the non-Fermi-liquid behavior. Here we investigate a possibility of having a superconducting off-diagonal long-range order (ODLRO) and a pseudogap phase within the SYK framework. We found that ODLRO may be established in spin-1/2 version of the model with the time-reversal invariance and an extra attractive interaction. If the latter is taken as the on-site negative-$U$ Hubbard term, it leads to the pseudogap phase at $U<U_c$ dominated by quantum fluctuations of local phases. These fluctuations are described by a quantum version of the Kuramoto model, traditionally employed to illustrate synchronization of classical non-linear oscillators. In the opposite limit of large $U$, the SYK+Hubbard model is approaching a certain generalization of the integrable Richardson model. We present exact diagonalization studies, along with analytic solutions of the aforementioned limiting cases. We also discuss possible holographic interpretations of the model, ODLRO and the pseudogap.
Highlights
The Sachdev-Ye-Kitaev (SYK) model [1,2] has received a great deal of attention in recent years as an exactly solvable model with non-Fermi-liquid properties [3,4,5,6,7,8,9]
The finite-temperature behavior of the SYK + Hubbard model is illustrated in Fig. 5, where we present the color plot of the logarithm of off-diagonal long-range order (ODLRO) on the temperature versus U/J plane
Following the earlier studies [39,40,41,42,43], we found that the spin-full version of the SYK model with extra attractive interactions may exhibit ODLRO and superconductivity
Summary
The Sachdev-Ye-Kitaev (SYK) model [1,2] has received a great deal of attention in recent years as an exactly solvable model with non-Fermi-liquid properties [3,4,5,6,7,8,9]. As phonons [39,40], pair hopping [42], or special correlations between matrix elements [43] Such upgraded SYK-like models exhibit superconducting correlations, which may be treated within the large-N mean-field approach. To detect superconductivity numerically in a finite-size system, we employ the notion of the off-diagonal long-range order (ODLRO) [59,60] It allows for a sharp definition of the condensate fraction and its dependence on temperature and the attraction strength for a large but finite N (number of sites). VII briefly summarizes our findings and lists some open problems
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